Official solution
"Suman is brilliant and dishonest" an be expressed as : \[{P \wedge \sim R}\]
So ``Suman is brilliant and dishonest if and only if Suman is rich'' can be expressed as :
\[\left( {P \wedge \sim R} \right) \leftrightarrow Q\]
Now negation of this = \[ \sim \left[ {\left( {P \wedge \sim R} \right) \leftrightarrow Q} \right]\]
You should know, \[p \leftrightarrow q \equiv q \leftrightarrow p\]
So, \[ \sim \left[ {\left( {P \wedge \sim R} \right) \leftrightarrow Q} \right] \equiv \sim \left[ {Q \leftrightarrow \left( {P \wedge \sim R} \right)} \right]\]